On Critical Values of Polynomials with Real Critical Points

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz ³.     

Rational Functions with Prescribed Critical Points

A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into the other by postcomposition with a linear-fractional transformation. We give an explicit formula for the number of classes having a given degree d and given multiplicities m₁,..., m_n of given n critical points, for generic positions of the critical points. This number is the multiplicity of the irreducible sl₂ representation with highest weight in the tensor product of the irreducible sl₂ representations with highest weights The classes are labeled by the orbits of critical points of a remarkable symmetric function which first appeared in the XIX century in studies of Fuchsian differential equations, and then in the XX century in the theory of KZ equations.     

Connections between Size Functions and Critical Points

A mathematical approach to the concept of shape of a submanifold ℳ︁ of a Euclidean space had previously been given by means of ‘measuring functions’ (e.g. diameter or volume) and of the derived ‘size functions’. This paper relates the study and the computation of any such size function to the structure of critical points of the associated measuring function.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.     

Deformations of Functions and F-Manifolds

We study deformations of functions on isolated singularities.A unified proof of the equality of Milnor and Tjurina numbersfor functions on isolated complete intersections singularitiesand space curves is given. As a consequence, the base spaceof their miniversal deformations is endowed with the structureof an F-manifold, and we can prove a conjecture of V. Goryunov,stating that the critical values of the miniversal unfoldingof a function on a space curve are generically local coordinateson the base space of the deformation. 2000 Mathematics SubjectClassification 32S05.     

Non Degeneracy of Critical Points of the Robin Function with Respect to Deformations of the Domain

We show a result of genericity for non degenerate critical points of the Robin function with respect to deformations of the domain.     

On Functions Whose All Critical Points Are Contained in a Ball

The normal decomposition of operator spaces into inductive scales of locally convex spaces in accordance with the classification of operators by their normal indices is considered. The canonical isomorphisms of operator spaces over Banach space are generalized to operators in locally convex spaces.     

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into R ^n+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L ^n/2 Ricci curvature norm in R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.     

Deformations of Real Rational Dynamics in Tropical Geometry

In this paper we study analytic properties of orbits given by real rational functions. We introduce some comparison methods which allow us to compare the real rational dynamics with automata given by (max, +) functions, passing through a kind of scale transform in tropical geometry. Such a scale transform gives a one-to-one correspondence of presentations between automata and real rational functions. We study invariant properties of the real rational dynamics under change of presentations of automata.     

Triangulations of Surfaces with Boundary and the Homotopy Principle for Functions without Critical Points

We consider triangulations of surfaces with boundary and marked points. These triangulations are classified with respect to flip equivalence. The results obtained are applied to the homotopy classification of functions without critical points on 2-manifolds. It is shown that the set of such functions satisfies the one-parametric h-principle.     

Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

Categorical Geometry and Integration Without Points

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419–489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical σ-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function μ with values in [0,1] can be extended to a measure on an abstract σ-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.     

Integer Points of Entire Functions

A result is proved which implies the following conjecture ofOsgood and Yang from 1976: if f and g are non-constant entirefunctions, such that T(r, f) = O(T(r,g)) as r and such thatg(z Z implies that f(z) Z, then there exists a polynomialG with coefficients in Q, such that G(Z) Z and f = G g. 2000Mathematics Subject Classification 30D20, 30D35.